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In coordinate geometry radical axis is defined as locus for which power is same with respect to two circles.

If we take two points as centre of two circles. Now without disturbing centre of circle if I increase radius of one circle will radical axis move away from it or get closer to it and vice versa.

Also if there is one big(radius) circle and another small(radius) circle. Is radical axis closer to centre of bigger(radius) circle or smaller(radius) circle.

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  • $\begingroup$ Maybe you can start from something like $R^{{2}}=d_{{1}}^{{2}}-r_{{1}}^{{2}}=d_{{2}}^{{2}}-r_{{2}}^{{2}}$. From a picture on the relevant Wiki page, it's close to the smaller circle. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Sep 28 '18 at 15:38
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Denote $A$ and $B$ the centers, and $R$ and $r$ the radii of the bigger and the smaller circle, respectively.

Second question

Radical axis $\mathcal{P}$ is orthogonal to $AB$, and is closer to the bigger circle. This follows immediately from $D^2-R^2=d^2-r^2,$ which is equivalent to $$D^2-d^2=R^2-r^2.\quad\quad\quad(1)$$ Here $D$ denotes the distance from a point on $\mathcal{P}$ to $A,$ and $d$ the distance from this same point to $B.$

First question

For the point $E$ common to $AB$ and $\mathcal{P},$ we have $D+d=|AB|,$ from where $$D^2-d^2=|AB|\,\left(|AB|-2d\right)$$ and $$D^2-d^2=|AB|\,\left(2D-|AB|\right)$$ Putting together with (1) we obtain $$|AB|\,\left(|AB|-2d\right)=R^2-r^2=|AB|\,\left(2D-|AB|\right).$$ If $A,B$ do not move and $r$ is constant, then increasing $R$ implies decreasing $d$ and increasing $D.$ That is, $\mathcal{P}$ gets closer to the CENTER of the smaller circle and moves away from the center of the bigger circle.

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